We construct infinitely many Einstein-Weyl structures on S 2 × R of signature (− + +) which is sufficiently close to the model case of constant curvature, and whose space-like geodesics are all closed. Such structures are obtained from small perturbations of the diagonal of CP 1 × CP 1 using the method of LeBrun-Mason type twistor theory. The geometry of constructed Einstein-Weyl space is well understood from the configuration of holomorphic discs. We also review Einstein-Weyl structures and their properties in the former half of this article.Then f is C l for (λ, t) and C k−1 for z, and we obtain D (λ,t) = {f(λ, t; z) ∈ Z | z ∈ D}.Constructing similar map for each neighborhood of CP 1 × R, and patching them, we obtain the double fibration(7.6) (X + ,X R ) ̟ { { w w w w w w w w wf % % J J J J J J J J Ĵ M (Z, N ) (7.7)

A global twistor correspondence is established for neutral self-dual conformal structures with α-surface foliation when the structure is close to the standard structure on S 2 × S 2 . We need to introduce some singularity for the α-surface foliation such that the leaves intersect on a fixed two sphere. In this correspondence, we prove that a natural double fibration is induced on some quotient spaces which is equal to the standard double fibration for the standard Zoll projective structure. We also give a local general forms of neutral self-dual metrics with α-surface foliation.

Abstract. We construct examples of singular self-dual Zollfrei metrics explicitly, by patching a pair of Petean's self-dual split-signature metrics. We prove that there is a natural one-to-one correspondence between these singular metrics and a certain set of embeddings of RP 3 to CP 3 which has one singular point. This embedding corresponds to an odd function on R that is rapidly decreasing and pure imaginary valued. The one-to-one correspondence is explicitly given by using the Radon transform.

Abstract. In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by N. Hitchin. As geometric objects naturally associated to Einstein-Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein-Weyl manifold. Moreover we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.